Autocatalytic loop, amplification and diffusion: a mathematical and computational model of cell polarization in neural chemotaxis.

Causin P, Facchetti G - PLoS Comput. Biol. (2009)

Bottom Line:
We analyze further crosslinked effects and, among others, the contribution to polarization of internal enzymatic reactions, which entail the production of molecules with a one-to-more factor.The model shows that the enzymatic efficiency of such reactions must overcome a threshold in order to give rise to a sufficient amplification, another fundamental precursory step for obtaining polarization.Eventually, we address the characteristic behavior of the attraction/repulsion of axons subjected to the same cue, providing a quantitative indicator of the parameters which more critically determine this nontrivial chemotactic response.

ABSTRACTThe chemotactic response of cells to graded fields of chemical cues is a complex process that requires the coordination of several intracellular activities. Fundamental steps to obtain a front vs. back differentiation in the cell are the localized distribution of internal molecules and the amplification of the external signal. The goal of this work is to develop a mathematical and computational model for the quantitative study of such phenomena in the context of axon chemotactic pathfinding in neural development. In order to perform turning decisions, axons develop front-back polarization in their distal structure, the growth cone. Starting from the recent experimental findings of the biased redistribution of receptors on the growth cone membrane, driven by the interaction with the cytoskeleton, we propose a model to investigate the significance of this process. Our main contribution is to quantitatively demonstrate that the autocatalytic loop involving receptors, cytoplasmic species and cytoskeleton is adequate to give rise to the chemotactic behavior of neural cells. We assess the fact that spatial bias in receptors is a precursory key event for chemotactic response, establishing the necessity of a tight link between upstream gradient sensing and downstream cytoskeleton dynamics. We analyze further crosslinked effects and, among others, the contribution to polarization of internal enzymatic reactions, which entail the production of molecules with a one-to-more factor. The model shows that the enzymatic efficiency of such reactions must overcome a threshold in order to give rise to a sufficient amplification, another fundamental precursory step for obtaining polarization. Eventually, we address the characteristic behavior of the attraction/repulsion of axons subjected to the same cue, providing a quantitative indicator of the parameters which more critically determine this nontrivial chemotactic response.

pcbi-1000479-g011: Spectral analysis of the model.Eigenvalues (in absolute value) of the Jacobian matrix of the DCC simplified model at steady state, as a function of the angular coordinate (for symmetry represented here only in the range ).

Mentions:
We use the model to study the time scales that characterize the process on the front and back sides, respectively. To perform this mathematical analysis, we consider a simplified version of the DCC model, neglecting diffusion and feedback terms. By doing so, we yield a system of ordinary differential equations, decoupled sector by sector. In this study, we prescribe a-priori an asymmetric receptor distribution to describe the polarized situation reached after a sufficient time of exposure to the cue. In particular, we start from the steady state distribution of receptors obtained from the simulation of the DCC model with . We compute in each sector the eigenvalues of the Jacobian matrix of the system in correspondence to its steady state. All the eigenvalues are real negative, indicating that the steady state is an attractive point. Based on the principal component of the corresponding eigenvector, we associate a chemical species with each eigenvalue. Then, using standard tracking techniques [41], we follow the variation of each eigenvalue along the GC perimeter. In Fig. 11, we plot the modulus of the eigenvalue associated with each species as a function of the angle . The eigenvalue associated with the slowest process on the front side () appears to be connected to PKA, while on the back side () it appears to be connected to . Observe that all the eigenvalues undergo a variation along the angle, even if for some of them this is not apparent in the logarithmic scale, required to appreciate the different relative behaviors. The graph shows the strong variation of the eigenvalue connected with . Moreover, the general trend of reduction of the absolute values passing from the front side to the back side indicates that the front dynamics is faster than the back one.

pcbi-1000479-g011: Spectral analysis of the model.Eigenvalues (in absolute value) of the Jacobian matrix of the DCC simplified model at steady state, as a function of the angular coordinate (for symmetry represented here only in the range ).

Mentions:
We use the model to study the time scales that characterize the process on the front and back sides, respectively. To perform this mathematical analysis, we consider a simplified version of the DCC model, neglecting diffusion and feedback terms. By doing so, we yield a system of ordinary differential equations, decoupled sector by sector. In this study, we prescribe a-priori an asymmetric receptor distribution to describe the polarized situation reached after a sufficient time of exposure to the cue. In particular, we start from the steady state distribution of receptors obtained from the simulation of the DCC model with . We compute in each sector the eigenvalues of the Jacobian matrix of the system in correspondence to its steady state. All the eigenvalues are real negative, indicating that the steady state is an attractive point. Based on the principal component of the corresponding eigenvector, we associate a chemical species with each eigenvalue. Then, using standard tracking techniques [41], we follow the variation of each eigenvalue along the GC perimeter. In Fig. 11, we plot the modulus of the eigenvalue associated with each species as a function of the angle . The eigenvalue associated with the slowest process on the front side () appears to be connected to PKA, while on the back side () it appears to be connected to . Observe that all the eigenvalues undergo a variation along the angle, even if for some of them this is not apparent in the logarithmic scale, required to appreciate the different relative behaviors. The graph shows the strong variation of the eigenvalue connected with . Moreover, the general trend of reduction of the absolute values passing from the front side to the back side indicates that the front dynamics is faster than the back one.

Bottom Line:
We analyze further crosslinked effects and, among others, the contribution to polarization of internal enzymatic reactions, which entail the production of molecules with a one-to-more factor.The model shows that the enzymatic efficiency of such reactions must overcome a threshold in order to give rise to a sufficient amplification, another fundamental precursory step for obtaining polarization.Eventually, we address the characteristic behavior of the attraction/repulsion of axons subjected to the same cue, providing a quantitative indicator of the parameters which more critically determine this nontrivial chemotactic response.

ABSTRACTThe chemotactic response of cells to graded fields of chemical cues is a complex process that requires the coordination of several intracellular activities. Fundamental steps to obtain a front vs. back differentiation in the cell are the localized distribution of internal molecules and the amplification of the external signal. The goal of this work is to develop a mathematical and computational model for the quantitative study of such phenomena in the context of axon chemotactic pathfinding in neural development. In order to perform turning decisions, axons develop front-back polarization in their distal structure, the growth cone. Starting from the recent experimental findings of the biased redistribution of receptors on the growth cone membrane, driven by the interaction with the cytoskeleton, we propose a model to investigate the significance of this process. Our main contribution is to quantitatively demonstrate that the autocatalytic loop involving receptors, cytoplasmic species and cytoskeleton is adequate to give rise to the chemotactic behavior of neural cells. We assess the fact that spatial bias in receptors is a precursory key event for chemotactic response, establishing the necessity of a tight link between upstream gradient sensing and downstream cytoskeleton dynamics. We analyze further crosslinked effects and, among others, the contribution to polarization of internal enzymatic reactions, which entail the production of molecules with a one-to-more factor. The model shows that the enzymatic efficiency of such reactions must overcome a threshold in order to give rise to a sufficient amplification, another fundamental precursory step for obtaining polarization. Eventually, we address the characteristic behavior of the attraction/repulsion of axons subjected to the same cue, providing a quantitative indicator of the parameters which more critically determine this nontrivial chemotactic response.